5,682 research outputs found
Phase Retrieval From Binary Measurements
We consider the problem of signal reconstruction from quadratic measurements
that are encoded as +1 or -1 depending on whether they exceed a predetermined
positive threshold or not. Binary measurements are fast to acquire and
inexpensive in terms of hardware. We formulate the problem of signal
reconstruction using a consistency criterion, wherein one seeks to find a
signal that is in agreement with the measurements. To enforce consistency, we
construct a convex cost using a one-sided quadratic penalty and minimize it
using an iterative accelerated projected gradient-descent (APGD) technique. The
PGD scheme reduces the cost function in each iteration, whereas incorporating
momentum into PGD, notwithstanding the lack of such a descent property,
exhibits faster convergence than PGD empirically. We refer to the resulting
algorithm as binary phase retrieval (BPR). Considering additive white noise
contamination prior to quantization, we also derive the Cramer-Rao Bound (CRB)
for the binary encoding model. Experimental results demonstrate that the BPR
algorithm yields a signal-to- reconstruction error ratio (SRER) of
approximately 25 dB in the absence of noise. In the presence of noise prior to
quantization, the SRER is within 2 to 3 dB of the CRB
Bispectrum Inversion with Application to Multireference Alignment
We consider the problem of estimating a signal from noisy
circularly-translated versions of itself, called multireference alignment
(MRA). One natural approach to MRA could be to estimate the shifts of the
observations first, and infer the signal by aligning and averaging the data. In
contrast, we consider a method based on estimating the signal directly, using
features of the signal that are invariant under translations. Specifically, we
estimate the power spectrum and the bispectrum of the signal from the
observations. Under mild assumptions, these invariant features contain enough
information to infer the signal. In particular, the bispectrum can be used to
estimate the Fourier phases. To this end, we propose and analyze a few
algorithms. Our main methods consist of non-convex optimization over the smooth
manifold of phases. Empirically, in the absence of noise, these non-convex
algorithms appear to converge to the target signal with random initialization.
The algorithms are also robust to noise. We then suggest three additional
methods. These methods are based on frequency marching, semidefinite relaxation
and integer programming. The first two methods provably recover the phases
exactly in the absence of noise. In the high noise level regime, the invariant
features approach for MRA results in stable estimation if the number of
measurements scales like the cube of the noise variance, which is the
information-theoretic rate. Additionally, it requires only one pass over the
data which is important at low signal-to-noise ratio when the number of
observations must be large
Beyond Low Rank + Sparse: Multi-scale Low Rank Matrix Decomposition
We present a natural generalization of the recent low rank + sparse matrix
decomposition and consider the decomposition of matrices into components of
multiple scales. Such decomposition is well motivated in practice as data
matrices often exhibit local correlations in multiple scales. Concretely, we
propose a multi-scale low rank modeling that represents a data matrix as a sum
of block-wise low rank matrices with increasing scales of block sizes. We then
consider the inverse problem of decomposing the data matrix into its
multi-scale low rank components and approach the problem via a convex
formulation. Theoretically, we show that under various incoherence conditions,
the convex program recovers the multi-scale low rank components \revised{either
exactly or approximately}. Practically, we provide guidance on selecting the
regularization parameters and incorporate cycle spinning to reduce blocking
artifacts. Experimentally, we show that the multi-scale low rank decomposition
provides a more intuitive decomposition than conventional low rank methods and
demonstrate its effectiveness in four applications, including illumination
normalization for face images, motion separation for surveillance videos,
multi-scale modeling of the dynamic contrast enhanced magnetic resonance
imaging and collaborative filtering exploiting age information
Conjugate phase retrieval in a complex shift-invariant space
The conjugate phase retrieval problem concerns the determination of a
complex-valued function, up to a unimodular constant and conjugation, from its
magnitude observations. It can also be considered as a conjugate phaseless
sampling and reconstruction problem in an infinite dimensional space. In this
paper, we first characterize the conjugate phase retrieval from the point
evaluations in a shift-invariant space , where the generator
is a compactly supported real-valued function. If the generator
has some spanning property, we also show that a conjugate phase retrievable
function in can be reconstructed from its phaseless samples
taken on a discrete set with finite sampling density. With additional phaseless
measurements on the function derivative, for the B-spline generator of
order which does not have the spanning property, we find sets
and of cardinalities and respectively,
such that a conjugate phase retrievable function in the spline space
can be determined from its phaseless Hermite samples
, and . An
algorithm is proposed for the conjugate phase retrieval of piecewise
polynomials from the Hermite samples. Our results provide illustrative examples
of real conjugate phase retrievable frames for the complex finite dimensional
space \C^N
Sampling and Reconstruction of Spatial Signals
Digital processing of signals f may start from sampling on a discrete set Γ, f →(f(ϒη))ϒηεΓ. The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and sparse signals, and to nontraditional sampling methods, such as phaseless sampling. In this dissertation, we first consider the sampling and Galerkin reconstruction in a reproducing kernel space. The fidelity measure of perceptual signals, such as acoustic and visual signals, might not be well measured by least squares. In the first part of this dissertation, we introduce a fidelity measure depending on a given sampling scheme and propose a Galerkin method in Banach space setting for signal reconstruction. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-projection algorithm in a reproducing kernel subspace of Lp. A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. We introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. We split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have l_2 uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. We also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises. Phase retrieval (Phaseless Sampling and Reconstruction) arises in various fields of science and engineering. It consists of reconstructing a signal of interest from its magnitude measurements. Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. We consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We find an equivalence between nonseparability of signals in a shift-invariant space and their phase retrievability with phaseless samples taken on the whole Euclidean space. We also introduce an undirected graph to a signal and use connectivity of the graph to characterize the nonseparability of high-dimensional signals. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in shift-invariant spaces, that are determined by their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. We also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only
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